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| {{about|curvature of knots|the theorem concerning straight-line embeddings of planar graphs|Fáry's theorem}}
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| In the [[knot theory|mathematical theory of knots]], the '''Fary–Milnor theorem''', named after [[István Fáry]] and [[John Milnor]], states that three-dimensional [[smooth curve]]s with small [[total curvature]] must be [[unknot]]ted. The theorem was proved independently by Fáry in 1949 and Milnor in 1950. It was later shown to follow from the existence of [[quadrisecant]]s {{harv|Denne|2004}}.
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| ==Statement of the theorem== | |
| If ''K'' is any closed [[curve]] in [[Euclidean space]] that is sufficiently [[smooth curve|smooth]] to define the [[Curvature#Curvature of space curves|curvature]] κ at each of its points, and if the [[total curvature]] is less than or equal to 4π, then ''K'' is an [[unknot]], i.e.:
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| :<math> \text{If} \, \oint_K \! \kappa(s) \, \operatorname{d}s \le 4 \pi \ \text{then} \ K \ \text{is an unknot}. </math>
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| The [[contrapositive]] tells us that if ''K'' is not an unknot, i.e. ''K'' is not [[homotopy#Isotopy|isotopic]] to the circle, then the total curvature will be strictly greater than 4π. Notice that having the total curvature less than or equal to 4π is merely a [[sufficient condition]] for ''K'' to be an unknot; it is not a [[necessary condition]]. In other words, although all knots with total curvature less than or equal to 4π are the unknot, there exist unknots with curvature strictly greater than 4π.
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| ==Generalizations to non-smooth curves==
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| For closed polygonal chains the same result holds with the integral of curvature replaced by the sum of angles between adjacent segments of the chain. By approximating arbitrary curves by polygonal chains one may extend the definition of total curvature to larger classes of curves, within which the Fary–Milnor theorem also holds ({{harvnb|Milnor|1950}}, {{harvnb|Sullivan|2007}}).
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| ==References==
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| {{reflist}}
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| *{{citation
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| | last = Denne | first = Elizabeth Jane | |
| | arxiv = math/0510561
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| | publisher = University of Illinois at Urbana-Champaign
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| | series = Ph.D. thesis
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| | title = Alternating quadrisecants of knots
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| | year = 2004}}.
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| *{{citation|first=I.|last=Fary<!-- Sic. The original publication does not accent Fáry's name. -->|authorlink=István Fáry|url=http://www.numdam.org/item?id=BSMF_1949__77__128_0|title=Sur la courbure totale d’une courbe gauche faisant un nœud|journal=Bulletin de la Société Mathématique de France|volume=77|year=1949|pages=128–138}}.
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| *{{citation|doi=10.2307/1969467|first=J. W.|last=Milnor|authorlink=John Milnor|title=On the total curvature of knots|journal=[[Annals of Mathematics]]|volume=52|year=1950|issue=2|pages=248–257}}.
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| *{{cite arxiv|first=John M.|last=Sullivan|title=Curves of finite total curvature|year=2007|eprint=math/0606007 }}.
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| ==External links==
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| *{{citation|url=http://www.ics.uci.edu/~eppstein/junkyard/knot-curvature.html|title=The total curvature of a knot (long)|first=Stephen A.|last=Fenner|year=1990}}. Fenner describes a geometric proof of the theorem, and of the related theorem that any smooth closed curve has total curvature at least 2π.
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| {{DEFAULTSORT:Fary-Milnor theorem}}
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| [[Category:Knot theory]]
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| [[Category:Theorems in topology]]
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